In my lecture notes, I have found the notation $\|A\|_*$ for a matrix norm.
Do you know the name of this norm (such that I can read the definition of it), or do you even know the definition of it?
Thank you very much.
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Rodrigo de Azevedo
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user136457
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1Not an answer because I can't place the star notation, but take a look at the different norms at https://en.wikipedia.org/wiki/Matrix_norm to see if any of them seem likely. In particular, I would say the two most common are the Frobenius norm and operator norm, depending on the application. – Mario Carneiro Feb 20 '15 at 16:00
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Thank you for the link. However, this was also my first idea and I have already taken a look at the norms there. And I am sure that is neither the maximum norm, nor the Frobenius norm, nor the $p$-norm for any $p$, so I am a little bit stuck here. – user136457 Feb 20 '15 at 16:01
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2If you have anything in your note about the properties of the norm, that might be a useful hint in finding which one you mean. – Ben Grossmann Feb 20 '15 at 16:02
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Actually it is the first lecture and only provides an overview about the topics. This is why there is not much of context, but it has to do something with principal component pursuit, where we want to minimize $||L||_* + \lambda ||S||_1$ subject to $L + S = X$, where $X$ is a data matrix, $L$ a low-rank matrix and $S$ a sparse matrix. – user136457 Feb 20 '15 at 16:04
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Oh, I am so sorry. I see that it is on this webpage (after I read the answer to this post) but I could not find it! So sorry!! – user136457 Feb 20 '15 at 16:05
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This norm (according to conventional notations) is called the nuclear norm and is defined as $\|A\|_*=Tr(\sqrt{A^*A})$ where $A^*$ is the Hermitian conjugate of $A$.
Samrat Mukhopadhyay
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Thank you very much. I do not know why I did not see it when looking at the Matrix norm-Wikipedia page. So thank you for pointing that out, it helped a lot! – user136457 Feb 20 '15 at 16:06
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Its okay, this happens a lot to many people (at least to me):-) – Samrat Mukhopadhyay Feb 20 '15 at 16:07
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That looks like the definition of the Frobenius norm, not of the nuclear norm. – Rodrigo de Azevedo May 09 '18 at 09:36
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@RodrigodeAzevedo, please see this Wiki entry for the Nuclear norm. – Samrat Mukhopadhyay May 09 '18 at 10:39
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Oops, sorry for the misplaced square bracket. I have fixed it now. – Samrat Mukhopadhyay May 09 '18 at 10:53
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A similar asterisk notation is also used to represent the dual norm of a vector. See page 637 of Boyd & Vandenberghe's Convex Optimization. See also Dual norm intuition.
The OP specifies that this is a matrix norm, but just providing some references in case anyone is as confused as I was about the notation overload.
Rodrigo de Azevedo
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phntm
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